Selected Title s i n Thi s Serie s 48 Yoa v Benyamin i an d Jora m Lindenstrauss , Geometri c nonlinea r functiona l analysis , Volume 1 , 200 0 47 Yur i I . Manin , Frobeniu s manifolds , quantu m cohomology , an d modul i spaces , 199 9 46 J . Bourgain , Globa l solution s o f nonlinear Schrodinge r equations , 199 9 45 Nichola s M . Kat z an d Pete r Sarnak , Rando m matrices , Frobeniu s eigenvalues , an d monodromy, 199 9 44 Max-Alber t Knus , Alexande r Merkurjev , an d Marku s Rost , Th e boo k o f involutions, 199 8 43 Lui s A . Caffarell i an d Xavie r Cabre , Full y nonlinea r ellipti c equations , 199 5 42 Victo r Guillemi n an d Shlom o Sternberg , Variation s o n a theme b y Kepler , 199 0 41 Alfre d Tarsk i an d Steve n Givant , A formalization o f set theor y without variables , 198 7 40 R . H . Bing , Th e geometri c topolog y o f 3-manifolds , 198 3 39 N . Jacobson , Structur e an d representation s o f Jordan algebras , 196 8 38 O . Ore , Theor y o f graphs, 196 2 37 N . Jacobson , Structur e o f rings, 195 6 36 W . H . Gottschal k an d G . A . Hedlund , Topologica l dynamics , 195 5 35 A . C . Schaeffe r an d D . C . Spencer , Coefficien t region s fo r Schlich t functions , 195 0 34 J . L . Walsh , Th e locatio n o f critical point s o f analytic an d harmoni c functions , 195 0 33 J . F . Ritt , Differentia l algebra , 195 0 32 R . L . Wilder, Topolog y o f manifolds , 194 9 31 E . Hill e an d R . S . Phillips , Functiona l analysi s an d semigroups , 195 7 30 T . Rado , Lengt h an d area , 194 8 29 A . Weil , Foundation s o f algebrai c geometry , 194 6 28 G . T . Whyburn , Analyti c topology , 194 2 27 S . Lefschetz , Algebrai c topology , 194 2 26 N . Levinson , Ga p an d densit y theorems , 194 0 25 Garret t Birkhoff , Lattic e theory , 194 0 24 A . A . Albert , Structur e o f algebras, 193 9 23 G . Szego , Orthogona l polynomials , 193 9 22 C . N . Moore , Summabl e serie s an d convergenc e factors , 193 8 21 J . M . Thomas , Differentia l systems , 193 7 20 J . L . Walsh, Interpolatio n an d approximatio n b y rationa l function s i n the comple x domain, 193 5 19 R . E . A . C . Pale y an d N . Wiener , Fourie r transform s i n the comple x domain , 193 4 18 M . Morse , Th e calculu s o f variations i n the large , 193 4 17 J . M . Wedderburn , Lecture s o n matrices , 193 4 16 G . A . Bliss , Algebrai c functions , 193 3 15 M . H . Stone , Linea r transformation s i n Hilber t spac e an d thei r application s t o analysis , 1932 14 J . F . Ritt , Differentia l equation s fro m th e algebrai c standpoint , 193 2 13 R . L . Moore , Foundation s o f point se t theory , 193 2 12 S . Lefschetz , Topology , 193 0 11 D . Jackson , Th e theor y o f approximation, 193 0 10 A . B . Coble , Algebrai c geometr y an d thet a functions , 192 9 9 G . D . Birkhoff , Dynamica l systems , 192 7 8 L . P . Eisenhart , Non-Riemannia n geometry , 192 7 7 E . T . Bell , Algebrai c arithmetic , 192 7 (Continued in the back of this publication) This page intentionally left blank Geometri c Nonlinea r Functiona l Analysi s Volum e 1 This page intentionally left blank http://dx.doi.org/10.1090/coll/048 America n Mathematica l Societ y Colloquiu m Publication s Volum e 48 Geometri c Nonlinea r Functiona l Analysi s Volum e 1 Yoa v Benyamin i Jora m Lindenstraus s America n Mathematica l Societ y Providence , Rhod e Islan d Editorial Boar d Joan S . Birma n Susan J . Priedlander , Chai r Stephen Lichtenbau m 1991 Mathematics Subject Classification. Primar y 46-XX ; Secondary 22-XX , 28-XX , 47-XX , 52-XX , 54-XX . ABSTRACT. Thi s volum e i s devoted t o the stud y o f uniforml y continuou s and , i n particular , Lip - schitz function s betwee n Banac h spaces . Als o studied i s the related questio n o f classifying Banac h spaces an d thei r subset s (mainl y spheres ) wit h respec t t o unifor m o r Lipschit z classification . Library o f Congres s Cataloging-in-Publicatio n Dat a Benyamini, Yoav , 1943 - Geometric nonlinea r functiona l analysi s / Yoa v Benyamini , Jora m Lindenstrauss . p. cm . — (America n Mathematical Societ y colloquium publications, ISS N 0065-9258 ; v. 48) Includes bibliographica l references . ISBN 0-8218-0835- 4 (hardcove r : alk . paper ) 1. Nonlinea r functiona l analysis . I . Lindenstrauss , Joram , 1936 - . II . Title . III . Series : Colloquium publication s (America n Mathematica l Society ) ; v. 48 . QA321.5.B46 200 0 515'.7-dc21 99-1773 4 CIP Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission should be addressed to the Assistant to the Publisher, America n Mathematical Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionQams.org. © 200 0 b y the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e at URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 5 0 4 0 3 0 2 0 1 0 0 Dedicated t o A . Dvoretzky, our teacher , colleagu e an d frien d Founder o f the functiona l analysi s schoo l i n Israe l This page intentionally left blank Contents Introduction 1 1. Th e Subjec t Matte r 1 2. Th e Book . Conventions , Prerequisites , Etc . 7 3. Acknowledgement s 9 Chapter 1 . Retractions , Extension s an d Selection s 1 1 1. Absolut e Lipschit z Retract s 1 1 2. Extensio n o f Maps on Hilbert Spac e 1 8 3. Michael' s Selectio n Theore m 2 1 4. Lipschit z an d Unifor m Selection s 2 6 5. Note s an d Remark s 3 1 Chapter 2 . Retractions , Extension s an d Selection s (Specia l Topics ) 3 5 1. Approximatio n o f Uniformly Continuou s Function s 3 5 2. Th e Neares t Poin t Ma p 4 0 3. Th e Contractio n Extensio n Propert y 4 4 4. Th e Steine r Poin t 4 8 5. Simultaneousl y Continuou s Map s 5 3 6. Note s an d Remark s 5 8 Chapter 3 . Fixe d Point s ' 6 1 1. Continuou s Map s 6 1 2. Lipschit z Map s 6 3 3. Nonexpansiv e Map s 6 5 4. Approximatio n o f Fixed Point s 7 0 5. Note s an d Remark s 7 9 Chapter 4 . Differentiatio n o f Conve x Functions 8 3 1. Basi c Definition s an d Result s 8 3 2. Conve x Functions 8 5 3. Note s an d Remark s 9 6 Chapter 5 . Th e Radon-Nikody m Propert y 9 9 1. Vecto r Measure s an d Integratio n o f Vector-Valued Function s 9 9 2. Th e Radon-Nikody m Propert y 10 2 3. Differentiability , Tree s and the RNP 11 0 4. Example s Relate d t o the RNP 11 4 5. Note s an d Remark s 12 1 Chapter 6 . Negligibl e Set s and Gateau x Differentiabilit y 12 5 1. Haa r Nul l Set s 12 5 2. Gaussia n Measure s 13 5 x CONTENT S 3. Gaus s Nul l Set s 14 1 4. Gateau x Differentiabilit y o f Lipschitz Function s 15 3 5. Example s Relate d t o Freche t Differentiabilit y 15 6 6.